Preconditioning for PDE-constrained Optimization

For many partial differential equations (PDEs) and systems of PDEs efficient preconditioning strategies now exist which enable effective iterative solution of the linear or linearised equations which result from discretization. Few would now argue for the use of Krylov subspace methods for most such PDE problems without effective preconditioners.

There are many situations, however, where it is not the solution of a PDE which is actually required, but rather the PDE expresses a physical constraint in an Optimization problem: a typical example would be in the context of the design of aerodynamic structures where PDEs describe the external flow around a body whilst the goal is to select its shape to minimize drag.

We will briefly describe the mathematical structure of such PDE-constrained Optimization problems and show how they lead to large scale saddle-point systems. The main focus of the talk will be to present new approaches to preconditioning for such Optimization problems. We will show numerical results in particular for problems where the PDE constraints are provided by the Stokes equations for incompressible viscous flow.